AP Physics - Kinetic Theory of Matter
Kinetic Theory of Matter:
We’ve been using the kinetic theory of matter to explain thermal energy and the transfer of heat from one system to
another.
To put this down in a formal way, one can write down the key concepts about the kinetic theory. The kinetic theory can be
used for all states of matter, but for some reason, the theory seems to concentrate on gases.
These are the main assumptions for the theory. You should be familiar with them. Again, this is for a gas.
1.
2.
3.
4.
5.
The number of particles in a system is enormous and the separation between the particles is huge.
All the particles move randomly.
The particles have perfectly elastic collisions with each other and other atoms.
There are no forces of attraction between the particles of a gas.
All the particles are identical.
One of the things you laboriously mastered in chemistry was the good old mole. Remember the mole?
Here is the definition of a mole:
Mole  Avogadro’s number of a thing.
Okay, so what is Avogadro’s number? Well, that’s the number of things in a mole, like 12 is the number of things in a dozen.
Avogadro ' s Nr
 NA
 6.02 x 1023 thingees
A mole, like a dozen, is a counting unit and is of very little use for most things. For example, you would not want to go to
Burger King and order a mole of french fries. If they actually had enough french fries to make the sale, you would have so
many french fries that you could probably cover the surface of the earth with them to a pretty good depth.
This is because a mole is a really big, huge, vast, enormous, gargantuan number.
Moles are very useful, however, when we want to deal with large numbers of very small things. That is why you made use
of moles in chemistry for atoms and molecules.
That is what we will do in AP Physics as well.
The number of moles present in a sample is quite easy to calculate:
n
N
NA
n is the number of moles, N is the number of particles in the system, and NA is Avogadro’s number.
The properties of a gas can be described by the ideal gas law. This is one of the real staples of chemistry.
PV  nRT
P is pressure, V is volume, n is the number of moles, R is the universal gas constant, and T is the temperature in
Kelvins.
You will be provided with this equation on the AP Test.
There are two values for the universal gas constant that we will make use of. We need different values of R for when we use
different units. Change the units you use in the equation and you change the value of R that you need to use.
R  8.31
J
mol  K
R  0.0821
Use for the SI system
L  atm
mol  K
This is the one you probably used in chemistry.
mol is the abbreviation for a mole.
You probably remember three gas laws from chemistry as well; Boyle’s law, Charles’ law, and Gay-Lussac’s law. These
laws described the behavior of a system that was undergoing change. In Boyle’s law, the pressure and volume change while
the temperature stays constant. Charles’ law deals with volume and temperature. Gay-Lussac’s law is about pressure and
temperature. All three of them can be put together in the combined gas law.
PV
PV
1 1
 2 2
T1
T2
PV
1 1  P2V2
If the temperature doesn’t change, then you get Boyle’s law:
And so on.
You will not be provided with any of these laws, but they are easy to develop from the ideal gas law. We solve the ideal gas
law for R, the universal gas constant.
PV  nRT
R
PV
nT
Now we look a system that is undergoing a change – temperature, pressure, or volume. We have a set of initial conditions
and a set of final conditions:
R
PV
1 1
nT1
and
R
P2V2
nT2
Both expressions equal R, so we can set them equal to each other.
PV
PV
1 1
 2 2
nT1 nT2
The number of moles hasn’t changed, so the n term cancels and we have the combined gas law.
PV
PV
1 1
 2 2
T1
T2
Energy of the Particles: Temperature is a direct measure of the kinetic energy of the particles. We can find the average kinetic
energy of a particle by using the following equation:
3
K avg  k BT
2
Kavg is the average kinetic energy per particle for the gas, kB is known as Boltzmann’s constant, and T is the
temperature in Kelvins. This would be the amount of kinetic energy for a single particle.
The value of Boltzmann’s constant is:
kB  1.38 x 1023
J
K
You will be provided with this equation (and the value of the different constants) on the AP Test.
Dear Doctor Science,
How many pins fit on the head of an angel?
-- Carol Janacek from Brooklyn, NY
Dr. Science responds:
Avogadro's Number. That's 6 point something times 10 to the 23rd, I believe. Avogadro, of course, was
both a scientist and a bishop of the Holy Roman Empire. He and Thomas Aquinas captured Visigoths
and made them dress as any one of the nine choirs of angels. The didactic duo then drove crude pins -or spikes, as we would call them today -- into the unlucky Barbarians. Then, as moles devoured the
Visigoths, researchers would count up the number of spikes being used and arrived at Avogadro's
Number. How this experiment ended up revealing accurate data about angel's heads is beyond this
modern day scientist, but then there's a lot in the history of science that defies an upbeat explanation. For
good reasons, I guess. I suppose there's some sort of poetic justice to this whole story. If so, it eludes me.

What is the average kinetic energy per molecule of a tank of oxygen gas at 22.0 C?
3
K avg  k BT
2
TK  TC  273.15
TK  22.0  273.15  295.15 K
3
J 
K avg  1.38 x 1023   295.15 K   611 x 1023 J
2
K 
K avg 
6.11 x 1021 J
Botzmann’s constant is given by:
kB 
R
NA
Where kB is Boltzmann’s constant, R is the universal gas constant, and NA is Avogardro’s number.
To find the total kinetic energy of the gas, you just need to multiply the average kinetic energy per particle by the number of
particles.
3
E  NkBT
2
Total kinetic energy of system.
E is the total kinetic energy of the system, N is the number of particles in the system, kB is Boltzmann’s constant, and T
is the temperature in Kelvins.
We also know that Boltzmann’s constant is given by:
kB 
R
NA
Plug this into the total energy equation and we get:
E
N
NA
But
3
Nk BT
2

3  R 
N
T
2  NA 
is simply the number of moles in the system, so the equation becomes:
3
E  nRT
2
Total kinetic energy of a gas system.
You will not be provided with this equation. Be prepared to derive the thing.

Find the total energy in a tank that has 2.50 mol of helium at 25.0 C.
T = 298.15 K
E
3
E  nRT
2
3
J

 2.50 mol   8.31
 298.15 K  
2
mol  K 

9 290 J
Pressure of a Gas: We can explain the behavior of gases and the origin of pressure using the kinetic theorem. You need to
understand how the theorem does this.
Within the system there is a huge number of particles smashing into each other and into the walls of the container. These
collisions transfer energy and exert forces on the particles. Each time a gas molecule smashes into the container wall, it
pushes it outward. It exerts a force on the wall. Add all these forces up and you have an average force that is fairly constant
over time. Divide the average force by the area and you get the pressure.
The collisions are responsible for the gases’ pressure.
When you blow up a balloon, it gets larger. Why?
Let’s look at a partially inflated balloon. Inside the balloon, the air molecules are busy smashing into the interior surface of
the balloon, pushing it out. But there are air molecules on the outside of the balloon having collisions with it, pushing it in.
An equilibrium develops in which the force pushing in on the balloon is equal to the force pushing out on the balloon. So
the balloon stays the same size.
Now you blow additional air into the balloon. More air molecules mean more collisions with the inside surface of the
balloon, so the balloon is pushed outward and the balloon get bigger. Eventually the balloon is big enough so that there are
more collisions with air molecules on the outside. Eventually the force from the outside collisions becomes equal to the
force from the collisions on the inside. A new equilibrium is reached and the balloon has a new size.
What happens when you increase the temperature of the system?
The particles gain kinetic energy, the collisions with the container are more violent, so they push harder. The force
increases, therefore the pressure increases. If the system cools, just he opposite happens.
Average Velocity of a Particle: The average velocity of a particle is given by this equation:
vrms 
3RT
3k BT

M
m
vrms is the root mean square velocity (figure average) of the particle, R is the universal gas constant, T is
the temperature in Kelvins, M is the molecular mass (mass for one mole of an element or compound), kB is
Boltzmann’s constant, and m is the mass of a single particle.
You will be provided this equation on the AP Test.

What is the average velocity of the particles of oxygen at 255 K?
We’ll use this equation:
vrms 
3k BT
m
First we need mass of an oxygen molecule.

  32.0 g
1 mol
1 molecule 

23
 6.02 x 10 molecules   1 mol
vrms
vrms
2

23 kg  m
31.38 x 10
255 K 
2 


K

s

 
5.316 x 1026 kg


m2
 19.86 x 10 2
s
4

446
  1 kg
  3
  10 g

26
  5.316 x 10 kg

m2
 198.6 x 10 2
s
3
m
s
Dear Doctor Science,
Why is the hot on the left and the cold on the right?
-- Richard O'Donnell from Colorado Springs, CO
Dr. Science responds:
There are exceptions to this rule. If the plumber who installed the fixture is handicapped by left-handedness, then hot
and cold could be reversed. Certain megalomaniacal plumbers install the figures backwards just to give vent to their
sociopathic compulsions to scald others. In Spanish speaking countries, like Mexico or Southern California, C stands
for "Caliente", and F, of course, for "Freezing", or "Freakout". My Spanish is a little rusty, but you get the idea.
Dear Doctor Science,
How do my cats know the outdoor temp is warm enough to go out? I never see them reading the
thermometer, and I don't allow them to watch the news. It only upsets them.
-- Joe DeMuth from Munci, Indiana
Dr. Science responds:
You're wise to shield your cats from the nightly news. Cats have been known to become psychotic when exposed to
Dan Rather. Felines determine the outdoor temperature from the level of irritation they sense in you when you're
stuck holding the door open for them. Sure, sometimes they make a show of sniffing the outdoor air, but that's just
a ruse to make you think they're being objective when they're simply being codependent and manipulative. Fact
is, cats hate the outdoors, and only venture outside when their litter box needs to be changed.
Dear Cecil:
If cold is simply the absence of heat, i.e., the absence of rapidly moving molecules of water or air, then how come vacuum-packed
canned food doesn't come out frozen, or at least very cold? And then if you walked a hundred feet out of your spaceship with a
glass of water, would the water freeze because of the vacuum, or would it boil since there's no air pressure or barometric pressure
to overcome?
--Barry H., Chicago
Cecil replies:
Christ Almighty, Barry, you're asking for a short course in thermodynamics. Don't you guys want to know about Neil Sedaka
anymore?
Let's clear up a couple misconceptions to start with. First, your idea that cold is "the absence of rapidly moving molecules of
water or air" is a bit confused. Cold refers to very slow-moving molecules of anything, whether water, air, or Eskimo Pies. If
you have no molecules at all, the concept of temperature is meaningless. That's why it's technically incorrect to speak of the
"cold of outer space"--strictly speaking, space has no temperature, period. (On the other hand, space will make objects that
are floating around in it cold--in some cases, very cold. Space is what's known as a "temperature sink," meaning it sucks
heat out of things. But we'll get back to this in a minute.)
Second, a vacuum never causes water to freeze; it causes water to boil. As air pressure decreases, so does boiling point.
That's why water boils much faster on a mountaintop than it does at sea level. By the same token, you can make water boil
at room temperature in the laboratory by applying a partial vacuum.
Now then. The contents of an earthbound vacuum-packed can do not freeze because they're in contact with the sides
thereof--they absorb room heat by conduction. There is no room heat in space, though, so the temperature of a solid object
floating in the void consists of the difference between the heat the object absorbs from the sun and the internal heat it
radiates away. This temperature is dependent on such things as the reflectance of the object's surface, its shape, mass,
orientation toward the sun, and so on.
Polished aluminum will absorb sufficient heat to raise its temperature as high as 850 degrees Fahrenheit; certain types of
white paint, on the other hand, absorb so little heat that their temperature may not get much above -40 Fahrenheit, even in
full sunlight. Parts of the space shuttle get down to -180 to -250 degrees Fahrenheit.
Theoretically, the temperature of an object in deep space could get down pretty close to absolute zero, -460 degrees
Fahrenheit. But even in the middle of nowhere there's enough in the way of stray particles and radiation to heat thing up to
3 degrees Kelvin--that is, the equivalent of 3 degrees Celsius (5 degrees F) above absolute zero.
Finally, we have the question of liquids in space. In a vacuum most liquids have such a low boiling point that they vaporize
almost instantly. For that reason, most substances exist in space in either the gaseous or the solid state. When the
astronauts take a leak while on a mission and expel the result into space, it boils violently. The vapor then passes
immediately into the solid state (a process known as desublimation [or deposition]), and you end up with a cloud of very
fine crystals of frozen tinkle. It is by such humble demonstrations that great scientific truths are conveyed.